# On the Bhattacharya-Mesner rank of third order hypermatrices

## TL;DR

This paper introduces the Bhattacharya-Mesner rank for third order hypermatrices, providing bounds, conditions for inverses, and extending classical linear algebra theorems to hypermatrices.

## Contribution

It defines a new hypermatrix rank, establishes bounds, and generalizes key linear algebra concepts like invertibility and the rank-nullity theorem.

## Key findings

- Defined Bhattacharya-Mesner rank for hypermatrices
- Derived bounds for tensor rank using this new rank
- Extended matrix inverse and rank-nullity theorem to hypermatrices

## Abstract

We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the connection relating the rank to a notion of linear dependence. We also derive explicit necessary and sufficient conditions for the existence of third order hypermatrix inverse pair. Finally we use inverse pair to extend to third order hypermatrices the formulation and proof of the matrix rank-nullity theorem.